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In the book Open Problems in Topology by Jan Van Mill and George M. Reed, the following problem was presented: 108. Is there a para- Lindelof Dowker space? Recall that a para-Lindelof Dowker space has a locally countable open refinement, satisfies Axiom T4, and is not countably paracompact. Some results on this problem are in http://topology.auburn.edu/tp/reprints/v11/tp11203.pdf, where it is shown that the conditions are preserved under perfect mappings. What is the status of this problem? Any references are appreciated.

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  • $\begingroup$ What does it mean for a space to have "a locally countable open refinement"$\:$? $\;\;$ $\endgroup$
    – user5810
    Aug 6, 2012 at 22:33
  • $\begingroup$ @Ricky If a space has a locally countable open refinement, then it locally has the same cardinality of the natural numbers in a refinement $V$ (a cover such that for every $v\inV$ there exists $u\inU$ such that $V\subsetU$). $\endgroup$
    – user22393
    Aug 6, 2012 at 22:43
  • $\begingroup$ .....and what does it mean for two sets to have "locally the same cardinality" ("in a refinement" or otherwise)? $\endgroup$ Aug 6, 2012 at 23:20
  • $\begingroup$ Steven's first question is a good question. $\;\;$ Whatever the answer to that is, what you have in parentheses seems to be equivalent to $\: V\subseteq U \:$. $\;\;\;\;$ $\endgroup$
    – user5810
    Aug 6, 2012 at 23:40
  • $\begingroup$ Given that the question makes no sense and the OP has made no attempt to fix it, I'm having trouble understanding why it's still open. $\endgroup$ Aug 7, 2012 at 12:06

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It is now problem 502 in "Open problems in topology II", I would guess it is still open.

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